I am remarkably impressed with the multilevel abilities currently available in Mplus. However, I was wondering if you might suggest a citation that describes exactly how Mplus is estimating a particular type of multilevel model.

specifically, I've used two child-level predictors & one child-level outcome from the high school & beyond study. Using the %within% and %between% statements in Mplus, I can fully replicate the fixed and random effects for both intercepts and slopes that are obtained in PROC MIXED.

here is my question: specifically how is Mplus estimating these values? My understanding is that the between & within approach of say Muthen (1994) clearly only allows for random intercepts whether under FIML or MUML. How are random slopes possible here?

again, i'm tremendously pleased with these capabilities, I'm just trying to understand this estimation process analytically. any recommended readings would be most welcome.

Thanks for your comments. Mplus 2.1 introduced FIML estimation of models that go beyond the random intercepts models that MUML considered. Unlike MUML, FIML does not perform an analysis of between and within covariance matrices. Instead, full maximum-likelihood estimation with random slope models draws on raw data as in regular multilevel modeling using HLM, MLwiN, or PROC MIXED. I am currently preparing a paper together with Tihomir Asparouhov that shows how Mplus integrates such multilevel modeling with SEM features using a general latent variable framework, where latent variables can be random intercepts, random slopes, or factors. The estimation builds on the EM algorithm where random effects are viewed as missing data. The paper will shortly be placed on the Mplus web site in pdf form and it will be announced on SEMNET and Multilevelnet.

Thank you for your helpful reply. I was hoping I could ask just one more quick question. When reading McDonald & Goldstein (1989) and Muthen (1991), it seems to be the case that the FIML approach to multilevel SEM under unbalanced designs is analytically intractable, and this is what in part motivated the clever MUML approximation.

Would it be accurate to say that Mplus has been able to overcome the intractability of FIML in the unbalanced case for multilevel SEM? If so, in your opinion, does this functionally "solve" the problems encountered when violating iid assumptions in the entire class of SEM? For example, if the entire OLS regression model is nested within standard SEM, is the entire random effects regression now nested within multilevel SEM?

I suspect all of these questions and many more will be addressed in your upcoming paper, but I was hoping you could send a few lines with your thoughts as I wrestle with these interesting new developments.

FIML for the multilevel SEM was not intractable, just a bit more cumbersome computationally, and not directly implementable in conventional SEM. Hence the value of MUML. Both the FIML and the MUML approach was presented in Muthen (1990) - see Mplus web site under References/Multilevel Data. Essentially, instead of analyzing one within and one between group as in MUML, FIML would use one within group and D between groups, where D is the number of distinct cluster sizes.

The MUML model itself, however, is more limited than the models that the current Mplus FIML handles. MUML only does random intercept models, whereas FIML also allows random slopes for observed covariates just like in multilevel regression. So, the current FIML is more general than the random intercept version I discussed in 1990.

So, yes, random effects regression is now nested within "multilevel SEM" - although I would prefer to refer to it as "multilevel latent variable modeling" since this spans a broader framework.

We are still a bit away from being able to do general random slope modeling in SEM, however, because we don't yet allow random slopes for latent variables. This will be in the next Mplus version due out next year. But I would say that already now, we certainly take good care of the lack of independent observations through the current allowance of random intercepts and slopes.

It seems that random slopes model for observed variables can be handled using Mplus2.14, but the random path analysis model (both effects a-->b and b--->c were random) still not supported by MPLUS, right? Also, random slopes model for latent variables do not supported. Whether the new coming version can be used to handle those questions? Thanks.

I am currently running two-level analyses like example 9.1 (except my group level moderators also have within group variation). At the between level I have found that the moderator significantly predicts variation in the slope between the predictor and the criterion (i.e., s on w is significant). I was hoping to probe for simple slopes at high, medium, and low levels of the moderator using an Aiken and West type of procedure because all of my variables are continuous. Do you have any recommendations how to do this what the input might look like? Thank you.

I receive the following error message when predicting random slopes between two level one variables:

THE ESTIMATED BETWEEN COVARIANCE MATRIX IS NOT POSITIVE DEFINITE AS IT SHOULD BE. COMPUTATION COULD NOT BE COMPLETED. THE VARIANCE OF S1 APPROACHES 0. FIX THIS VARIANCE AND THE CORRESPONDING COVARIANCES TO 0, DECREASE THE MINIMUM VARIANCE, OR SPECIFY THE VARIABLE AS A WITHIN VARIABLE.

Is there any way to tell if constraining the variance is inappropriate?

I am trying to estimate the following random intercept regression model in Mplus 5:

Level 1: Yij = b0j + b1j(Xij) + rij

Level 2: boj = g00 + u0j b1j = g10

where boj is a random intercept, b1j = g10 is the fixed slope for the regression of Yij on the level 1 predictor Xij, rij is the level 1 residual, g00 is the average intercept across clusters, and u0j is a level 2 residual for the intercept.

My Mplus Model statement is as follows:

%WITHIN% Yij on Xij;

At first glance, the output looks like what I'm looking for, but I'm puzzled by the finding that the estimate for g00 (under "MEANS") is very different from what I get in HLM. Also, the estimate for g00 does not change when Xij is centered. I wonder what Mplus is actually estimating on the between level in this model? Is it the grand mean of the variable Xij or the average intercept across clusters (as I had expected)?

Funny enough, things are different when I run a random coefficient regression analysis in which *both* the intercept b0j and the slope b1j are random coefficients that vary across level-2-units:

B1j | Yij on Xij;

Then centering Xij makes a difference for the estimation of g00 as I would expect.

To compare to HLM, you need to put the covariate on the WITHIN list. Please see Examples 9.1 and 9.2 in the most recent user's guide for a description of what happens when the covariate is not on the WITHIN list.

in order to familiarize myself with Mplus I compared the output of model with a random intercept and random slopes with the output of the same model in HLM. Most estimates/standard errors differ only in the second/third decimal place but the significance of the random slopes differs clearly. I read that the HLM-Chi-Square test is one-sided, but even when I divide the Mplus-p-value by two the difference is still obvious:

Here are some details about my analyses/models: - I specified a model with two L1 covariates. The only L1 covariate is the mean of one of the L1 covariates. The L2 covariate is used to explain the intercept b0. - Both L1 covariates are on the within-list of Mplus. - I use a student level weight in both analyses. - I use MLR in Mplus and FIML in HLM. - I have 5 multiply imputed datasets. - The L1 covariates are z-standardized and group mean centered – whereas I find the same pattern without group mean centering.

Could you give me any hint, why all the other values differ by just a small amount and concerning the significance of the random slopes I find such a substantial difference?

A statistical review of our ms said that we should use a random regression model rather than a growth curve model BECAUSE we only had 3 time points. Our (continuous) data is non-linear, and GCM estimates random intercepts and random slopes. I am confused about the difference between RRM & GCM. Is there a clear advantage of RRM over GCM? Is there a good ref that compares RRM to GCM? Thank you, Peter

3 time points is not really enough for non-linear growth modeling. I don't know what is meant by random regression modeling. Could it be using splines? Random coefficient regression is another matter. Generalized additive models is yet another. Does somebody else know?

When you estimate a model with type= complex twolevel random and include one random slope, the output provides the variance for that random slope, as well as a standard error and p-value. What significance test is being applied here? Is this useful in the decision about whether the slope should be random or fixed? Raudenbush and Bryk descibe a chi square test as part of the decision process, but I can't tell if that is what MPlus is providing or not. thanks!

The significance test is the same simple z test as is provided for all parameters. This is not a great test for variances as is well-known because of testing at the boundary of the admissible parameter space. Many articles have been written about alternatives using a mixture of chi-squares, but may not be suitable in all the kinds of models that Mplus offers.

Hello Dr. Muthen, Please correct me if I am wrong in my logic and coding as I am not getting desired results. Consider: x--->y, m1 and m2 are moderators on relationship between x and y. there are 5 groups and each group is selecting its own y based on m1 and m2. I am using random coefficient regression for this and code is:-

Your random coefficient regression looks correct. It implies an interaction between x and m1, m2. If you have problems with this run, send input, output, data, and license number to support@statmodel.com.

But I don't see where your groups come in. If you have different y's for the 5 different groups, I would think you'd do 5 different analyses.